Inverse Fourier transform of $frac{sin(aomega)}{(omega-a)^2}^2$.
$begingroup$
The exercise asks that we find the IFT of
$$frac{sin(aomega)}{(omega-a)^2}^2$$
Using the general formula for IFT yields integrals of the type:
$$intlimits_{-infty}^inftyfrac{cos(aomega)}{(omega-a)^2}$$ which has no apparent solution (at the known integrals tables i checked).
I suspect some of FT's properties can be of use (multiplication on frequency domain = convolution in time domain) but since i cant find a function whose FT is in terms of $sin(aomega)$ or $(omega-a)^2$ I can't figure it out.
fourier-transform
edited Jan 11 at 20:32
rtybase
10.7k21533
asked Jan 11 at 17:45
ManouilManouil
93
$endgroup$
|
show 2 more comments
$begingroup$
The exercise asks that we find the IFT of
$$frac{sin(aomega)}{(omega-a)^2}^2$$
Using the general formula for IFT yields integrals of the type:
$$intlimits_{-infty}^inftyfrac{cos(aomega)}{(omega-a)^2}$$ which has no apparent solution (at the known integrals tables i checked).
I suspect some of FT's properties can be of use (multiplication on frequency domain = convolution in time domain) but since i cant find a function whose FT is in terms of $sin(aomega)$ or $(omega-a)^2$ I can't figure it out.
fourier-transform
edited Jan 11 at 20:32
rtybase
10.7k21533
asked Jan 11 at 17:45
ManouilManouil
93
$endgroup$
$begingroup$
The code for $-infty$ is just -infty :)
$endgroup$
– Greg Martin
Jan 11 at 18:16
$begingroup$
@GregMartin It doesn't display correctly (i copy pasted -infty in the place of infty)
$endgroup$
– Manouil
Jan 11 at 18:22
$begingroup$
I've already edited it. The "trick" is to use curly brackets to enclose all that you want as a subscript. (The same aplies for superscripts, numerators, denominators, etc.)
$endgroup$
– Alejandro Nasif Salum
Jan 11 at 18:35
1
$begingroup$
What is the FT of $e^{-a|t|}$ ? So what is the FT of $frac{1}{(omega-a)^2}$ ? So what is the FT of $frac{e^{i b omega}}{(omega-a)^2}$ and $frac{sin(a omega)^2}{(omega-a)^2}$ ?
$endgroup$
– reuns
Jan 11 at 18:38
$begingroup$
@reuns i believe FT of $e^{-a|t|}$ is $frac{2jomega}{a^2-omega^2}$
$endgroup$
– Manouil
Jan 11 at 19:06
|
show 2 more comments
$begingroup$
The exercise asks that we find the IFT of
$$frac{sin(aomega)}{(omega-a)^2}^2$$
Using the general formula for IFT yields integrals of the type:
$$intlimits_{-infty}^inftyfrac{cos(aomega)}{(omega-a)^2}$$ which has no apparent solution (at the known integrals tables i checked).
I suspect some of FT's properties can be of use (multiplication on frequency domain = convolution in time domain) but since i cant find a function whose FT is in terms of $sin(aomega)$ or $(omega-a)^2$ I can't figure it out.
fourier-transform
edited Jan 11 at 20:32
rtybase
10.7k21533
asked Jan 11 at 17:45
ManouilManouil
93
$endgroup$
The exercise asks that we find the IFT of
$$frac{sin(aomega)}{(omega-a)^2}^2$$
Using the general formula for IFT yields integrals of the type:
$$intlimits_{-infty}^inftyfrac{cos(aomega)}{(omega-a)^2}$$ which has no apparent solution (at the known integrals tables i checked).
I suspect some of FT's properties can be of use (multiplication on frequency domain = convolution in time domain) but since i cant find a function whose FT is in terms of $sin(aomega)$ or $(omega-a)^2$ I can't figure it out.
fourier-transform
fourier-transform
edited Jan 11 at 20:32
rtybase
10.7k21533
asked Jan 11 at 17:45
ManouilManouil
93
edited Jan 11 at 20:32
rtybase
10.7k21533
asked Jan 11 at 17:45
ManouilManouil
93
edited Jan 11 at 20:32
rtybase
10.7k21533
edited Jan 11 at 20:32
rtybase
10.7k21533
edited Jan 11 at 20:32
rtybase
10.7k21533
10.7k21533
asked Jan 11 at 17:45
ManouilManouil
93
asked Jan 11 at 17:45
ManouilManouil
93
asked Jan 11 at 17:45
ManouilManouil
93
93
$begingroup$
The code for $-infty$ is just -infty :)
$endgroup$
– Greg Martin
Jan 11 at 18:16
$begingroup$
@GregMartin It doesn't display correctly (i copy pasted -infty in the place of infty)
$endgroup$
– Manouil
Jan 11 at 18:22
$begingroup$
I've already edited it. The "trick" is to use curly brackets to enclose all that you want as a subscript. (The same aplies for superscripts, numerators, denominators, etc.)
$endgroup$
– Alejandro Nasif Salum
Jan 11 at 18:35
1
$begingroup$
What is the FT of $e^{-a|t|}$ ? So what is the FT of $frac{1}{(omega-a)^2}$ ? So what is the FT of $frac{e^{i b omega}}{(omega-a)^2}$ and $frac{sin(a omega)^2}{(omega-a)^2}$ ?
$endgroup$
– reuns
Jan 11 at 18:38
$begingroup$
@reuns i believe FT of $e^{-a|t|}$ is $frac{2jomega}{a^2-omega^2}$
$endgroup$
– Manouil
Jan 11 at 19:06
|
show 2 more comments
$begingroup$
The code for $-infty$ is just -infty :)
$endgroup$
– Greg Martin
Jan 11 at 18:16
$begingroup$
@GregMartin It doesn't display correctly (i copy pasted -infty in the place of infty)
$endgroup$
– Manouil
Jan 11 at 18:22
$begingroup$
I've already edited it. The "trick" is to use curly brackets to enclose all that you want as a subscript. (The same aplies for superscripts, numerators, denominators, etc.)
$endgroup$
– Alejandro Nasif Salum
Jan 11 at 18:35
1
$begingroup$
What is the FT of $e^{-a|t|}$ ? So what is the FT of $frac{1}{(omega-a)^2}$ ? So what is the FT of $frac{e^{i b omega}}{(omega-a)^2}$ and $frac{sin(a omega)^2}{(omega-a)^2}$ ?
$endgroup$
– reuns
Jan 11 at 18:38
$begingroup$
@reuns i believe FT of $e^{-a|t|}$ is $frac{2jomega}{a^2-omega^2}$
$endgroup$
– Manouil
Jan 11 at 19:06
$begingroup$
The code for $-infty$ is just -infty :)
$endgroup$
– Greg Martin
Jan 11 at 18:16
$begingroup$
The code for $-infty$ is just -infty :)
$endgroup$
– Greg Martin
Jan 11 at 18:16
$begingroup$
@GregMartin It doesn't display correctly (i copy pasted -infty in the place of infty)
$endgroup$
– Manouil
Jan 11 at 18:22
$begingroup$
@GregMartin It doesn't display correctly (i copy pasted -infty in the place of infty)
$endgroup$
– Manouil
Jan 11 at 18:22
$begingroup$
I've already edited it. The "trick" is to use curly brackets to enclose all that you want as a subscript. (The same aplies for superscripts, numerators, denominators, etc.)
$endgroup$
– Alejandro Nasif Salum
Jan 11 at 18:35
$begingroup$
I've already edited it. The "trick" is to use curly brackets to enclose all that you want as a subscript. (The same aplies for superscripts, numerators, denominators, etc.)
$endgroup$
– Alejandro Nasif Salum
Jan 11 at 18:35
1
1
$begingroup$
What is the FT of $e^{-a|t|}$ ? So what is the FT of $frac{1}{(omega-a)^2}$ ? So what is the FT of $frac{e^{i b omega}}{(omega-a)^2}$ and $frac{sin(a omega)^2}{(omega-a)^2}$ ?
$endgroup$
– reuns
Jan 11 at 18:38
$begingroup$
What is the FT of $e^{-a|t|}$ ? So what is the FT of $frac{1}{(omega-a)^2}$ ? So what is the FT of $frac{e^{i b omega}}{(omega-a)^2}$ and $frac{sin(a omega)^2}{(omega-a)^2}$ ?
$endgroup$
– reuns
Jan 11 at 18:38
$begingroup$
@reuns i believe FT of $e^{-a|t|}$ is $frac{2jomega}{a^2-omega^2}$
$endgroup$
– Manouil
Jan 11 at 19:06
$begingroup$
@reuns i believe FT of $e^{-a|t|}$ is $frac{2jomega}{a^2-omega^2}$
$endgroup$
– Manouil
Jan 11 at 19:06
|
show 2 more comments
0
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$begingroup$
The code for $-infty$ is just -infty :)
$endgroup$
– Greg Martin
Jan 11 at 18:16
$begingroup$
@GregMartin It doesn't display correctly (i copy pasted -infty in the place of infty)
$endgroup$
– Manouil
Jan 11 at 18:22
$begingroup$
I've already edited it. The "trick" is to use curly brackets to enclose all that you want as a subscript. (The same aplies for superscripts, numerators, denominators, etc.)
$endgroup$
– Alejandro Nasif Salum
Jan 11 at 18:35
1
$begingroup$
What is the FT of $e^{-a|t|}$ ? So what is the FT of $frac{1}{(omega-a)^2}$ ? So what is the FT of $frac{e^{i b omega}}{(omega-a)^2}$ and $frac{sin(a omega)^2}{(omega-a)^2}$ ?
$endgroup$
– reuns
Jan 11 at 18:38
$begingroup$
@reuns i believe FT of $e^{-a|t|}$ is $frac{2jomega}{a^2-omega^2}$
$endgroup$
– Manouil
Jan 11 at 19:06